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The sum of the digits of a three-digit number is 12. The sum of the squares of the digits is 72. What is the smallest possible value of the three-digit number? | nvidia |
Evaluate the expression \[ \frac{a+2}{a+1} \cdot \frac{b-1}{b-2} \cdot \frac{c + 8}{c+6} , \] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero. | nvidia |
Find the radius of the circle that passes through the points $(1,7),$ $(4,2),$ and $(5,5),$ given that the center of the circle lies on the line $y=x.$ | nvidia |
Find the base $b$ such that the number of three-digit numbers with distinct digits that can be formed using the digits 0 through 9 is equal to the number of four-digit numbers with distinct digits that can be formed using the digits 0 through 9, when written in base $b$. | nvidia |
A complex number $z$ is given such that $\frac{z^2}{z^*}$ is a real number. Determine the argument of $z$, given that $\text{Im}(z) > 0$. | nvidia |
Point $P$ lies on the circle $(x-2)^2+(y-3)^2=25$ and is 5 units from the point $(1,1)$. Find the sum of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
Determine the smallest possible degree of the polynomial $p(x)$ such that the graph of the rational function $\frac{p(x)}{x^4 + 2x^3 + 1}$ has a slant asymptote. | nvidia |
Let $f(x)$ be a monic polynomial of degree 6, such that $$\lim_{x\to \infty}\frac{f(x)-x^6}{x^4}=0,$$and let there exist nonzero constants $a$ and $b$ such that
$f(a)=b$,
$f(2a)=-b$,
$f(3a)=b$, and
$f(4a)=-b$.
Find the limit $\ds\lim_{x\to -\infty} f(x)$. | nvidia |
The sum of the digits of the number $3^{2017}$ is divisible by $9$. Determine if it's also divisible by $18$. | nvidia |
The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? | nvidia |
Find the smallest positive integer that has exactly three distinct prime divisors and is not a multiple of any of these primes squared. | nvidia |
How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
| nvidia |
Let $x$, $y$, $z$, and $w$ be real numbers such that $|x-y|=1$, $|y-z|=2$, and $|z-w|=5$. What is the sum of all possible values of $|x-w|$? | nvidia |
Let $n$ be a positive integer such that $n^2 - 20n + 101 = q$ for some positive integer $q$. What is the largest possible value of $q$? | nvidia |
Find the number of complex numbers that satisfy the equation $|z-2+3i| \le 5$, where $z$ is a complex number in the form $a+bi$, and $a$ and $b$ are integers. | nvidia |
Find the value of $b$ such that the polynomial $x^4-2bx^3+3x^2+2bx-8$ is divisible by $x^2+4$. | nvidia |
Let $S$ be a set of positive integers with the following property: for every integer $x$ in $S$, there are $x$ integers less than $x$ in $S$. What is the largest element of $S$ if the average of the elements of $S$ is 2017? | nvidia |
What is the smallest positive integer that can be expressed as the sum of two different prime numbers, each greater than 20? | nvidia |
Find the sum of all integers divisible by 7 between 1 and 49 (inclusive). | nvidia |
Let $\mathbf{B}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{B}^2 = \mathbf{I}$, where $\mathbf{I}$ is the identity matrix. Find the number of possible values that $\det(\mathbf{B})$ can take. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $abc = 27.$ Find the minimum value of $a + 3b + 9c.$ | nvidia |
Let $x$, $y$, and $z$ be real numbers such that $x$ is the greatest value and $x$ is less than or equal to twice the value of $z$. Let $a$, $b$, and $c$ be real numbers such that $a\geq b$ and $a$ is less than or equal to twice the value of $c$. For what value of $y$ will we always have that $(x-y)(y-z)$ is greater tha... | nvidia |
Compute the product of the roots of the equation $x\sqrt{x} - 2x + 5\sqrt{x} - 3 = 0,$ given that all of the roots are real and nonnegative. | nvidia |
Let $a$, $b$, and $c$ be three positive real numbers such that $a+b+c=1$ and $a^3+b^3+c^3+a^4+b^4+c^4=2(a^2+b^2+c^2)$. Find $a$, $b$, and $c$. | nvidia |
Find the ordered pair $(a,b)$ of real numbers such that
\[\sqrt{5 + 2 \cos \theta} = a \sin \theta + b \cos \theta\]
holds for all $\theta \in \mathbb{R}$. | nvidia |
Find the sum of all values of $z$ that satisfy the equation $\frac{4}{2z^2+5z-2}+\frac{5}{z^2+6z+9}=1$. | nvidia |
Find the greatest real number $a$ such that the equation $y = \frac{x^2-5x}{2x-3}$ has no solution for $x$ when $x < a$. | nvidia |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-... | nvidia |
In a triangle, the measure of the largest angle is 20 degrees less than the sum of the measures of the other two angles, and one of the smaller angles is 10 degrees more than half the largest angle. Find the number of degrees in the measure of the largest angle. | nvidia |
Find the number of four-digit positive integers with distinct digits and the thousands digit being an odd number. | nvidia |
Find all positive integers $m$ for which there exists a positive integer $n$ such that the equation $$n\sin(m\cos(\pi nx))=x$$ has infinitely many real solutions. | nvidia |
Consider a right triangle with legs of lengths $x$ and $y$ and hypotenuse of length $z$, where $x,y,z$ are positive integers. The area of the triangle can be expressed as $\frac{xy}{2}$, and the perimeter can be expressed as $x+y+z$. Find the ordered pair $(x,y)$ for which the sum of the area and perimeter is equal to ... | nvidia |
Given that the complex number $z$ satisfies $2z-3\bar{z}=-2-30i$, find the value of $|z|^2$. | nvidia |
The diagonal of a cube with twice the surface area of a cube with edge length 8 inches can be expressed in the form $m\sqrt[2]{n}$ where $m$ and $n$ are positive integers and $n$ contains no perfect square factors. Compute $m+n$. | nvidia |
Find the maximum value of the expression $\sqrt{x+y+1}+\sqrt{4-x-y}$ for real $x$ and $y$ satisfying $x+y \leq 2$. | nvidia |
Find the number of ways to arrange 16 books on a bookshelf such that 5 of the books are red and the remaining 11 books are blue, and no two red books are adjacent. | nvidia |
Find the value of $\frac{1}{a} + \frac{1}{b}$, given that $a$ and $b$ are positive integers, and $\frac{1}{a}$ and $\frac{1}{b}$ are in the ratio $3:5$. | nvidia |
Determine the sum of all two-digit replacements for $ab$ such that the number ${ab{,}ab{,}ab}$ is divisible by 99. | nvidia |
Find all real numbers $x$ so that the sum $(x + i) + (x + 1)i + (x + 2)i^2$ is a real number. Enter all the solutions, separated by commas. | nvidia |
Let $n$ be the smallest positive integer such that $9n$ is a perfect cube and the sum of the digits of $9n$ is a multiple of 6. What is the smallest possible value of $n$ such that these conditions are satisfied? | nvidia |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia |
The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ has the property that
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_6 = 2b_4$ then find $\ds\f{b_7}{b_1}$. | nvidia |
The harmonic mean of two positive integers is the reciprocal of the average of their reciprocals. If the harmonic mean of $m$ and $n$ is $\frac mn$ , where $m>n>1$, determine the value of $m$ in terms of $n$. | nvidia |
Find the sum of all real numbers that are tripled when added to their squares. | nvidia |
Find the sum of all possible values of $x$ that satisfy the equation $\log_x (3x) = 2.$ | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that
\[x^2+y^2+z^2=3\sqrt[3]{xyz}.\]
If $T=x+y+z$, determine the maximum value of $T^3$.
[Your response should end here] | nvidia |
The sum of the digits of a three-digit number is 12. How many such numbers are there that are divisible by 3? | nvidia |
Let $f(x)$ be a function defined for all real numbers $x$. If $f(a)f(b)=ab(f(a)+f(b))$ for any two real numbers $a$ and $b$, then find all possible values of $f(4)$. | nvidia |
Determine the percentage of positive integers less than 100 that have an odd number of divisors. | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{10} = \cos x.\] | nvidia |
Determine all possible values of $b$ such that the inequality
\[|x^2 - 2bx + 2b| < 1\]
has exactly one real solution. | nvidia |
Find the sum of all positive integer values of $a$ such that $6x^2+13x+a=0$ has two distinct real roots. | nvidia |
If $(a,b)$ is a point on the graph of the function $f(x)$, determine a point that must be on the graph of $f(2x)$ and find the product of its coordinates. | nvidia |
Let $x$ be the number of distinct two-digit positive integers whose digits are either 1, 3, or 5. Determine the number of distinct two-digit positive integers that have a digit 2 and are obtained by interchanging the tens and units digits of the integers included in $x$. | nvidia |
A point is chosen at random on the number line between 0 and 2, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 2, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point, but less than t... | nvidia |
Let $FGHIJ$ be a convex pentagon with $FG \parallel IJ, GH \parallel IF, GJ \parallel FH, \angle FGH=135^\circ, FG=4, GH=6,$ and $IJ = 18.$ Given that the ratio between the area of triangle $FGH$ and the area of triangle $IGJ$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers, find $p+q.$ | nvidia |
Find the sum of all real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1-\frac{1}{1+\frac{x^2}{1+x}}}.$$ | nvidia |
If $x$ is a real number such that $x+\frac{1}{x} = 2\sqrt{2}$, find $x^{6} + \frac{1}{x^{6}}$. | nvidia |
Determine the number of integers that satisfy the inequality $|x^2 - 5x + 6| < 2$. | nvidia |
What is the largest positive integer $n$ such that $5^n$ divides $13^{51} - 1$? | nvidia |
Find the number of distinct triangles whose vertices are at points with nonnegative integer coordinates in the plane and lie on the graph of the equation $x^2+y^2=2010$, such that the area of each triangle formed by these points and the origin is an integer. | nvidia |
Prove that for any acute angle $\theta$, $\tan \theta > \sin \theta$. | nvidia |
Find the smallest three-digit number that is divisible by 7 and has the sum of its digits equal to 9. | nvidia |
Let \(P(x)\) be a polynomial with real coefficients, and let \(z\) be a non-real root of the equation \(P(x)=0\). The graph of the equation \(y=P(x)\) is symmetric with respect to the line that passes through the point \((z,0)\) and is perpendicular to the line that connects \(z\) with its conjugate \(\overline{z}\).
... | nvidia |
What is the largest integer that can be written in two different ways as the sum of three distinct integers? | nvidia |
Consider a function $g(x)$ that satisfies the equation
\[g(x+y) + 2x = 2xg(y) + g(x)\]for all real numbers $x$ and $y.$ If $g(2) = 3$ then compute $g(10).$ | nvidia |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | nvidia |
Find the sum of cubes of all roots of the equation
$$x^3-7x^2+3x-1=0$$
if it is known that the sum of the roots is equal to 7. | nvidia |
Given that $\sqrt{xy} = b^b$ and $\log_b (x^{\log_b y}) + \log_b (y^{\log_b x}) = 4b^4$, find the value of $b$ for which the system has a solution $(x,y)$ in real numbers, where $x$ and $y$ are restricted to be positive integers. | nvidia |
Find the $x$ that satisfies $\log_{x+1} 5 = \frac{1}{3}$, given that $x$ is a rational number. | nvidia |
Find the value of $x$ such that the point $(x,2)$ lies on the line passing through $(1,4)$ and $(-3,-2)$. | nvidia |
If $p$ and $q$ are positive integers such that $p$ has exactly $3$ divisors, and $q$ has exactly $p$ divisors, then what is the greatest possible value of $\gcd(p,q)$? | nvidia |
Consider a positive integer n to be a prime tail if there exists a positive integer m such that the decimal representation of the mth prime number ends with exactly n zeroes. How many positive integers less than 1000 are not prime tails? | nvidia |
Find the largest positive real number $a$ such that
\[|x|^a + |y|^a \ge \frac{|x+y|^a}{a}\]for all real numbers $x$ and $y.$ | nvidia |
The sum of the squares of two consecutive positive integers is 145. The smaller integer is half of the sum of its digits. What is the smaller integer? | nvidia |
Determine the number of integers (positive and negative) that are divisors of 36. | nvidia |
Let $S$ be the set of all pairs of integers $(a,b)$ such that $a$ and $b$ are relatively prime and $a+b$ is even. Define
$$F(x) = \sum_{(a,b) \in S} \frac{x^{a+b}}{a^2+b^2}.$$
Find the value of $\frac{F(1)}{F(-1)}$. | nvidia |
Find the number of integers between 500 and 600 that have three distinct digits in decreasing order. For example, 543 is such an integer. | nvidia |
Consider a nonconstant polynomial $Q(x)$ with real coefficients such that
\[Q(Q(x)) = Q(x)Q(x+1) + x.\]
Find the polynomial $Q(x)$. | nvidia |
Find the number of three-digit positive integers that are congruent to 2 (mod 5) and have a sum of digits that is a multiple of 3. | nvidia |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | nvidia |
Let $z$ be a complex number such that $\ds z^3+z^{-3}=2i$. Determine the value of $z^6+z^{-6}$. | nvidia |
Given that $a$, $b$, and $c$ are positive integers such that $\mathop{\text{lcm}}[a,b]=21$ and $\mathop{\text{lcm}}[a,c]=14$, what is the smallest possible value of $\mathop{\text{lcm}}[a,b,c]$? | nvidia |
Find the smallest positive integer n such that n, n+1, and n+2 are all divisible by different prime numbers. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=1$ and $|x-y|=|y-z|$. Determine the maximum value of $x$. | nvidia |
Find the largest five-digit number that is divisible by each of the five smallest prime numbers. | nvidia |
Let $x$ and $y$ be positive real numbers such that
$x^3 - 3x^2y + 3xy^2 - y^3 = 0.$
Find the largest possible value of $\frac{x}{y}$. | nvidia |
How many different polynomials can be formed with distinct integer roots of the form $a + ib$ such that the polynomial also has at least one real root that is an integer? | nvidia |
Let $M$ be the product of the first $50$ positive even integers. Find the largest integer $n$ such that $M$ is divisible by $2^n$. | nvidia |
Find the number of positive integers $x$ that satisfy the equation $2x^2 + 3x - 5 \equiv 0 \pmod{11}$ and are less than 500. | nvidia |
Find the value of $d$ such that the quadratic $x^2 + 2x + d$ has a root that is twice another root. | nvidia |
Find the greatest natural number $n$ such that $n\leq 2008$ and $(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]$ is a perfect square.
| nvidia |
Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$
| nvidia |
Let a, b, and c be three positive numbers such that a*b*c=27 and (1/a)+(1/b)+(1/c)=1.
Find the sum a+b+c. | nvidia |
What is the sum of all positive integers less than 100 that are cubes of perfect cubes? | nvidia |
A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$. | nvidia |
Let $a$ and $b$ be two positive integers. What is the smallest positive integer $n$ such that $a^n$ is divisible by $b$? | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x+y) = g(x)g(y)\]for all positive integers $x$ and $y.$ Given $g(4) = 12$ and $g(7) = 21,$ find $g(11).$ | nvidia |
In a right triangle, the length of the hypotenuse is 10 cm, and one leg is 6 cm. Find the length of the other leg, knowing that the area of the triangle is 24 square centimeters. | nvidia |
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